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Metamath Proof Explorer

Theorem cdlemedb

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. D represents s_2. (Contributed by NM, 20-Nov-2012)

Ref Expression
Hypotheses cdlemeda.l 
|- .<_ = ( le ` K )
cdlemeda.j 
|- .\/ = ( join ` K )
cdlemeda.m 
|- ./\ = ( meet ` K )
cdlemeda.a 
|- A = ( Atoms ` K )
cdlemeda.h 
|- H = ( LHyp ` K )
cdlemeda.d 
|- D = ( ( R .\/ S ) ./\ W )
cdlemedb.b 
|- B = ( Base ` K )
Assertion cdlemedb 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> D e. B )

Proof

Step Hyp Ref Expression
1 cdlemeda.l 
 |-  .<_ = ( le ` K )
2 cdlemeda.j 
 |-  .\/ = ( join ` K )
3 cdlemeda.m 
 |-  ./\ = ( meet ` K )
4 cdlemeda.a 
 |-  A = ( Atoms ` K )
5 cdlemeda.h 
 |-  H = ( LHyp ` K )
6 cdlemeda.d 
 |-  D = ( ( R .\/ S ) ./\ W )
7 cdlemedb.b 
 |-  B = ( Base ` K )
8 hllat 
 |-  ( K e. HL -> K e. Lat )
9 8 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
10 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> K e. HL )
11 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> R e. A )
12 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> S e. A )
13 7 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. B )
14 10 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> ( R .\/ S ) e. B )
15 7 5 lhpbase 
 |-  ( W e. H -> W e. B )
16 15 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> W e. B )
17 7 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. B /\ W e. B ) -> ( ( R .\/ S ) ./\ W ) e. B )
18 9 14 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> ( ( R .\/ S ) ./\ W ) e. B )
19 6 18 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> D e. B )